By definition, we have that \( \mathcal {FW}_n^k \subseteq \mathbb {S}^n_+ \)  for all \(k \in [n]\)  and \(\mathcal {FW}_n^2 = SDD_n\) , and \( \mathcal {FW}_n^n = \mathbb {S}^n_+ \)  [1]}. \( \mathcal {FW}_n^k\)  can be decomposed into a sum of PSD matrices of rank at most \(k\)  as follows: \(Z \in \mathcal {FW}_n^k\)  if and only if it can be written as \( Z = \sum _{i=1}^{\binom{n}{k}} V_i \Lambda _i V_i^T \) , where \(V_i\)  is an \(n \times k \)  matrix whose columns each contain one nonzero element which is equal to 1, and \(\Lambda _i \in \mathbb {S}^k_+\)  [2]}. One could transform an optimization problem over the PSD cone into an conic optimization problem of smaller dimension. However, enforcing this constraint is problematic as \(n\)  or \(k\)  increases, which grows in a combinatorial fashion \({\binom{n}{k}} \) . Therefore, using factor-width k matrices to approximate the PSD cone is not trivial in a practical way.
